Calculus 2 | Index

Calculus 2 goes deeper into integrals and then turns to infinite series.

The course covers integration techniques, improper integrals and more uses of integrals. It also covers differential equations, parametric and polar calculus, convergence tests, and power and Taylor series.

It’s for AP Calculus BC students and college Calculus II students.

What to know first: Calculus 1, especially u-substitution, the Fundamental Theorem of Calculus, and areas and volumes.

Lessons for this course are coming soon. The list below shows every planned lesson in teaching order.

Units in this course

Unit 1: Integration Techniques

  1. Integration by Parts (coming soon) — integrate products of functions, such as x times a sine or an exponential
  2. Trigonometric Integrals (coming soon) — integrate powers and products of sine, cosine, tangent and secant
  3. Trigonometric Substitution (coming soon) — integrate expressions with square roots of sums and differences of squares
  4. Integration by Partial Fractions (coming soon) — integrate rational functions by splitting them into simpler fractions
  5. Hyperbolic Functions (coming soon) — the definitions, identities, derivatives and integrals of sinh, cosh and tanh
  6. Choosing an Integration Technique (coming soon) — a strategy for deciding which method fits an integral
  7. Simpson’s Rule (coming soon) — approximate a definite integral with parabolas and compare it with other numerical methods

Also see: partial fraction decomposition in Precalculus, Unit 2, and trig identities in Trigonometry, Unit 6.

Unit 2: Improper Integrals

  1. Improper Integrals (coming soon) — integrals over infinite intervals and how to tell if they converge
  2. Improper Integrals with Infinite Discontinuities (coming soon) — integrals where the function blows up inside or at the edge of the interval
  3. Comparison Test for Improper Integrals (coming soon) — decide if an improper integral converges by comparing it with a known one

Unit 3: Applications of Integrals

  1. Shell Method (coming soon) — find the volume of a solid of revolution with cylindrical shells
  2. Arc Length of a Curve (coming soon) — find the length of a curve with an integral
  3. Surface Area of Revolution (coming soon) — find the area of the surface made by spinning a curve around an axis
  4. Work Done by a Variable Force (coming soon) — find the work done by springs, cables and other changing forces
  5. Pumping Liquid Problems (coming soon) — find the work needed to pump liquid out of a tank
  6. Hydrostatic Force (coming soon) — find the force of a fluid on a submerged plate
  7. Center of Mass (coming soon) — find moments and the centroid of a flat region

Also see: area between curves and the disk and washer methods in Calculus 1, Unit 9.

Unit 4: Differential Equations

  1. Verifying Solutions to Differential Equations (coming soon) — what a differential equation is and how to check a solution
  2. Slope Fields (coming soon) — draw and read slope fields to see what solutions look like
  3. Euler’s Method (coming soon) — approximate a solution step by step with tangent lines
  4. Separable Differential Equations (coming soon) — solve differential equations by separating the variables
  5. Exponential Models with Differential Equations (coming soon) — solve dy/dt = ky for growth and decay
  6. Newton’s Law of Cooling (coming soon) — model how an object’s temperature approaches the temperature around it
  7. Logistic Differential Equations (coming soon) — model growth that slows as it nears a carrying capacity

Also see: logistic functions in Precalculus, Unit 3, and the full Differential Equations course.

Unit 5: Parametric and Polar Calculus

  1. Derivatives of Parametric Equations (coming soon) — find dy/dx and the second derivative for a parametric curve
  2. Arc Length of Parametric Curves (coming soon) — find the length of a curve given by x(t) and y(t)
  3. Vector-Valued Functions (coming soon) — find velocity, speed and acceleration for motion in the plane
  4. Derivatives of Polar Functions (coming soon) — find the slope of a polar curve
  5. Area in Polar Coordinates (coming soon) — find the area inside a polar curve
  6. Area Between Polar Curves (coming soon) — find the area inside one polar curve and outside another
  7. Arc Length of Polar Curves (coming soon) — find the length of a polar curve

Also see: parametric equations in Precalculus, Unit 6, and polar coordinates in Trigonometry, Unit 9.

Unit 6: Sequences and Series Convergence

  1. Convergence of Sequences (coming soon) — decide whether a sequence has a limit and find it
  2. Infinite Series and Partial Sums (coming soon) — what it means for an infinite series to converge or diverge
  3. Divergence Test (coming soon) — use the nth term test to show a series diverges
  4. Telescoping Series (coming soon) — find the sum of a series whose terms cancel
  5. Integral Test (coming soon) — decide if a series converges by comparing it with an improper integral
  6. p-Series (coming soon) — know when the sum of 1 over n to the power p converges
  7. Direct Comparison Test (coming soon) — compare a series term by term with one you already know
  8. Limit Comparison Test (coming soon) — compare two series with the limit of the ratio of their terms
  9. Alternating Series Test (coming soon) — decide if a series with alternating signs converges
  10. Absolute and Conditional Convergence (coming soon) — tell absolutely convergent series from conditionally convergent ones
  11. Ratio Test (coming soon) — decide convergence from the ratio of consecutive terms
  12. Root Test (coming soon) — decide convergence from the nth root of the terms
  13. Choosing a Convergence Test (coming soon) — a strategy for picking the right test for a series

Also see: infinite geometric series in Algebra 2, Unit 8.

Unit 7: Power Series and Taylor Series

  1. Taylor Polynomials (coming soon) — approximate a function near a point with a polynomial
  2. Alternating Series Error Bound (coming soon) — bound the error when you stop an alternating series early
  3. Lagrange Error Bound (coming soon) — bound the error of a Taylor polynomial approximation
  4. Power Series (coming soon) — what a power series is and how to write a function as one with a geometric series
  5. Radius and Interval of Convergence (coming soon) — find where a power series converges
  6. Taylor Series (coming soon) — find the Taylor series of a function centered at any point
  7. Maclaurin Series (coming soon) — the Maclaurin series for e to the x, sine, cosine and 1/(1 − x), and new series made by substituting into them
  8. Differentiating and Integrating Power Series (coming soon) — find new series by working term by term
  9. Binomial Series (coming soon) — expand (1 + x) to the power k as an infinite series for any real number k

For Teachers: Standards Alignment

Lessons follow the AP Calculus BC topics that go beyond AB, such as integration by parts, improper integrals, Euler’s method, logistic models, parametric and polar calculus, and series. They also follow a typical second-semester college Calculus II course. The slope fields and separable equations from AP Calculus AB Unit 7 are taught here, in Unit 4. Some topics, such as trig substitution, surface area, work and hydrostatic force, are taught in college courses but not tested on the AP exam.

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